accretion of the terrestrial planets and the earth-moon system

Canup and Agnor: Accretion of the Terrestrial Planets 113

113

1. INTRODUCTION

In the planetesimal hypothesis, the growth of terrestrialplanets is the result of the process of collisional accumula-tion from initially small particles in the protoplanetary disk.The accretion process is typically described in terms of threestages of growth, which are distinguished by our basic un-derstanding of the relevant physical processes involved informing solid bodies in a particular size range. The firststage involves the formation of kilometer-sized planetesi-mals from an initial protoplanetary disk of gas and dust. Bythe end of this stage of growth (discussed in chapter byWard, 2000), planetesimals have reached sizes large enoughso that their dynamical evolution is determined primarilyby gravitational interactions with the central star and withother planetesimals, rather than by surface, electromagnetic,or sticking forces. The middle stage consists of the accu-mulation of a swarm of kilometer-sized planetesimals intolunar- to Mars-sized planetary embryos (see chapter byKortenkamp et al., 2000). Numerous works have demon-strated that in this stage, dynamical friction acts to reduceencounter velocities with the largest bodies, facilitating the“runaway” growth of ~1025–1027 g (M = 5.98 × 1027 g)planetary embryos in as little as 105 yr (e.g., Greenberg et al.,1978; Wetherill and Stewart, 1993). The last stage then con-sists of the formation of the final few planets via the colli-sion and merger of tens to hundreds of planetary embryos.Evolution during this period is thought to be driven by dis-tant interactions between the embryos, and requires a fewtimes ~108 yr (e.g., Wetherill, 1992).

In one of the first modern works to examine terrestrialplanet formation, Safronov (1969) proposed that planetsaccreted in radially confined feeding zones, in a relativelyquiescent manner through the accumulation of small bod-ies. Developments in the past two decades suggest that thegenerally localized, runaway stage persists only until bod-ies grow to the size of the Moon or Mars, leaving manyplanetary embryos throughout the terrestrial region. Such asystem of embryos is dynamically unstable on timescalesthat are short (~106 yr) compared to the age of the solarsystem, and highly energetic collisions between embryosthen occur to yield the four terrestrial planets. In this sce-nario, the characteristics of the final planets are determinedmainly by the specifics of the last few large impacts thateach planet experiences. The stochastic nature of the finalstage thus yields an inherent degree of uncertainty for theoutcome of accretion in any given system. Indeed, a widerange of possible planetary architectures is found to arisefrom even nearly identical initial conditions, suggestive ofthe great variety of terrestrial planet systems that might existin extrasolar systems.

The physical and dynamical environment in which theaccretion of the terrestrial planets took place is directly rel-evant to models of lunar formation. In the giant impact sce-nario, the Moon forms as a result of a single impact withEarth late in its formation history. While works to date havegenerally considered the various stages of planet accretionand the formation of the Moon separately, a more holisticapproach may be required. In particular, models of the pro-posed lunar-forming impact and the accretion of the Moon

Accretion of the Terrestrial Planets and the Earth-Moon System

Robin M. CanupSouthwest Research Institute

Craig B. AgnorUniversity of Colorado

Current models for the formation of the terrestrial planets suggest that the final stage ofplanetary accretion is characterized by collisions between tens to hundreds of lunar to Mars-sized planetary embryos. In this view, large impacts are an inevitable outcome as a system ofembryos destabilizes to yield the final few planets. One such impact is believed to be respon-sible for the origin of the Moon. Improvements in numerical methods have recently allowedfor the first direct orbit integrations of the final stage of accretion, which is believed to persistfor ~108 yr. The planetary systems produced by these simulations bear a general resemblenceto the terrestrial planets, but on average differ from our system in the final number of planets(fewer), their orbital spacings (wider) and their eccentricities and inclinations (larger). The dis-crepancy between these predictions and the nearly circular orbits of both Earth and Venus issignificant, and is likely a result of the approximations inherent to the late-stage accretion simu-lations performed to date. Results from these works further highlight the important role of sto-chastic impact events in determining final planetary characteristics. In particular, impacts capableof supplying the angular momentum of the Earth-Moon system are predicted to be common.

114 Origin of the Earth and Moon

have yet to identify a single impact that can yield the finalmasses of the Earth and Moon, together with the currentsystem angular momentum (Cameron and Canup, 1998;Cameron, 2000). However, recent terrestrial accretion stud-ies suggest that impacts subsequent to the lunar-formingevent may have contributed significantly to the final massand/or angular momentum of the Earth-Moon system, offer-ing a possible resolution to this dilemma (Agnor et al., 1999).

In this chapter, we review recent simulations of late-stageaccretion and discuss the successes and weaknesses of thesemodels (section 2). We then address issues especially rel-evant to the formation of the Moon via giant impact, includ-ing late-stage impact statistics and the potential role ofmultiple impacts in affecting planetary spin angular mo-menta (section 3). A brief discussion of open issues is in-cluded in section 4.

2. ACCRETION OF THETERRESTRIAL PLANETS

The environment in which the final accretion of terres-trial-type planets takes place is dependent upon the outcomeof the preceeding runaway growth stage. Midstage accre-tion models utilizing both statistical treatments that modelthe entire terrestrial region, and N-body simulations of run-away growth within a local radial zone of the disk, yieldqualitatively similar results: embryos with masses ~0.01–0.1M , occupying nearly circular, low-inclination orbitsafter 105–106 yr. The embryos have typical orbital spacingsof ∆a ~ 10 RHill , where RHill is the mutual Hill radius givenby

Ra a m m

MHill ≡ + +1 2 1 21/3

2 3(1)

where a1, a2, m1, and m2 are the semimajor axes and massesof adjacent embryos, and M is the mass of the Sun. A sys-tem of two bodies on initially circular orbits will be stableagainst mutual collision so long as ∆a > 3.5 RHill (e.g.,Gladman, 1993). However, multiplanet systems (or planetswith nonzero initial eccentricities) require larger separationsfor stability. Chambers et al. (1996) performed numericalintegrations of like-sized planets intially on circular orbitsand found a separation of ∆a ≈ 8–10 RHill provided stabil-ity for ~106–107 yr, in fair agreement with the predictedspacings from the midstage accretion simulations (e.g.,Weidenschilling et al., 1997). Ito and Tanikawa (1999) con-ducted numerical stability analyses of planetary embryos,including initial eccentricities and inclinations, a range ofembryo masses, and Jupiter and Saturn. They found muchshorter stability times of ~104–105 yr for a system of N =14 embryos with ∆a ≈ 8–10 RHill separations and initialvalues of ⟨e⟩1/2 = 2⟨i⟩1/2 = 0.005. The latter timescales areclose to those found by the late-stage N-body simulations(described below) that begin with similar initial eccentrici-ties. For comparison, the current terrestrial planets havemutual separations ranging from about 26 to 40 RHill .

The boundary between the middle and late stages is gen-erally believed to be representative of a dynamical transi-tion: from a stage in which growth is dominated by colli-sions with local material, to one in which distant interac-tions among the embryos lead to collisions on much longertimescales. Recent simulations that model embryo forma-tion in the full terrestrial zone (0.5–1.5 AU) find that 90%of the system mass is contained in a few tens of embryosafter about a million years, with the remaining ~10% of themass contained in a swarm of much smaller planetesimals(Weidenschilling et al., 1997). These simulations have in-cluded effects due to gas drag and a parameterization ofdistant perturbations between embryos, but to date have notincluded collisional fragmentation. At present, it is not clearto what extent the planetesimal swarm persists or is regen-erated via collisional erosion throughout the late stage. Itis often assumed that all the small material in the disk wouldbe rapidly swept up by the embryos, since the timescale forembryo formation (105–106 yr) is much shorter than thetimescale for the accumulation of the final planets (~2 ×108 yr). In this case, the dynamics of the final stage aregoverned solely by gravitational interactions among andcollisions between the large embryos. Interactions amongembryos initially on nearly circular orbits lead to eccentric-ity growth, and then to orbit crossing; once orbital isola-tion is overcome, the embryos collide and merge. The accu-mulation of embryos into larger bodies then proceeds untilthe secular orbital oscillations (primarily the eccentricities)of the remaining bodies are insufficient to allow bodies toencounter each other, and a few planets remain on well-separated orbits. Until recently, simulations of this final stagewere limited to statistical treatments due to the large num-ber of orbital times involved. However, numerical tech-niques now exist that allow for direct integration of systemsof N ~ 10–100 embryos for ~108 yr. Results from simula-tions using both methods are reviewed in the next section.

2.1. Late-Stage Simulations

Late-stage terrestrial accretion has been modeled usingtwo basic (and complementary) techniques. Monte Carlosimulations follow the orbital evolution of embryos in astatistical manner based on two-body scattering events (e.g.,Wetherill, 1985), while N-body orbital integrations directlytrack the trajectories of each embryo at all times (Cham-bers and Wetherill, 1998, hereafter CW98; Agnor et al.,1999, hereafter ACL99). Under comparable sets of assump-tions, both methods produce similar configurations of finalplanets.

Wetherill was the first to model the dynamical evolutionof systems of planetary embryos throughout the terrestrialregion. His Monte Carlo scheme, with its approximate treat-ment of dynamical interactions, is computationally fast andhas been used to generate large numbers of planetary sys-tems for a range of starting conditions and assumptions (e.g.,Wetherill, 1985, 1992, 1994, 1996). In this method, theprobability of each body experiencing a close encounter


with any other body is determined using an adaptation ofthe Opik (1951) formalism; the outcome of the encounteris then a function of a randomly selected distance of clos-est approach. Wetherill (1992) presented results of 434Monte Carlo simulations of the evolution of a few hundredembryos throughout the terrestrial and asteroid belt region.Fragmentation was assumed to occur when the impact en-ergy exceeded a chosen threshold, at which point the totalmass involved in a collision was divided into a small num-ber (i.e., 4) of equal-sized pieces. Bodies with massessmaller than 8 × 1025 g (or about the mass of the Moon)were removed from the simulation, in order to limit the totalnumber of objects to a computationally manageable level.Effects due to perturbations from and resonances with Ju-piter and Saturn were included in the form of simple para-meterizations. The planetary systems resulting from thesesimulations broadly resembled the current planets in num-ber and size, with some discrepancies. In particular, Earth-like planets with large eccentricities were common out-comes; of the final planets with masses greater than 3.5 ×1027 g (0.58 M ), ~40% had eccentricities greater than 0.05and ~75% had eccentricities larger that of Earth or Venus.

Direct integration methods explicitly track the gravita-tional interactions among all bodies in a simulation. How-ever, this increase in dynamical accuracy comes at the costof computational speed, which limits the number of simu-lations and bodies in a simulation that can be considered.Direct integrations of the late stage have recently becomepossible due to developments in symplectic integration tech-niques, which afford system energy and angular momen-tum conservation for the requisite ~108 orbits (e.g., Wisdomand Holman, 1991; Duncan et al., 1998; Chambers, 1998b).

CW98 performed 27 late-stage integrations that eachbegan with a system of roughly 25–50 embryos on circularorbits distributed throughout the terrestrial region (0.5–1.8 AU). All collisions were assumed to result in completeand inelastic merger. Figure 1 shows the evolution of a sys-tem of planetary embryos in mass and semimajor axis spacefor 100 m.y. (CW98, their Fig. 1). The first encounter be-tween embryos often occurs in the inner part of the disk,and results in a scattering of the two bodies involved suchthat their neighbors then become subject to close encoun-ters. This leads to what is described as a “wave” of closeencounters that travels outward through the terrestrial re-gion. Interactions among the embryos, including both reso-nances and scattering events, drive the accretion processuntil the final planets are so well-separated and few in num-ber that their mutual perturbations are insufficient to allowfor further collisions. Note that in the last frame of Fig. 1there are still crossing orbits, and so accretion is this caseis likely not yet complete.

CW98 also investigated the effect of Jupiter and Saturnon embryo accretion in the 0.55–4.0-AU region by addingthe giant planets (with their current masses and orbits) to asubset of their simulations after 107 yr (see Fig. 2). Theextent to which the giant planets were present and at theircurrent locations during the late stage is uncertain. Requir-

ing that Jupiter and Saturn formed before the nebula dis-persed implies a formation time of less than ~107 yr, sug-gesting that they were most likely present during most ofthe final accretion period. However, these planets may havemigrated significantly during and subsequent to their for-mation. In general, perturbations from outer giant planetsact to decrease the stability of a system of protoplanets (e.g.,Ito and Tanikawa, 1999), as both mean motion and secularresonances drive large-amplitude eccentricity oscillations.Of particular importance are the ν5 and the ν6 secular reso-nances, which occur where the apsidal precession rate oforbiting material is near to that of Jupiter or Saturn respec-tively (or at about 0.6 AU for the ν5 and 2.1 AU for the ν6,assuming current planetary orbital elements and no nebulargas disk).

The CW98 simulations indicate that planetary embryoswith orbital radii larger than ~1.2 AU are typically scatteredinto the ν6 or other, mean-motion resonances where theybecome dynamically coupled to the outer planets, leadingto either collisions with embryos in the terrestrial region orejection from the solar system following close encounterswith Jupiter. In their current positions, Jupiter and Saturnthus appear to prevent the accumulation of embryos intoterrestrial-sized planets for a > 1.2 AU. Given that the massof Mars is only ≈0.1 M , it is possible that Mars itself isa leftover planetary embryo formed via runaway growth(Wetherill, 1992; CW98). In the inner disk, the ν5 resonancecan lead to embryo removal via collisions with the Sun. Insimulations that include Jupiter and Saturn, at least 15% ofthe total initial embryo mass is typically lost through colli-sions with the Sun and hyperbolic ejection events (CW98).

ACL99 performed direct integrations of the late stage thatbegan with either 50 (m ~ 0.04 M) or 22 (m ~ 0.10 M)planetary embryos distributed between 0.5 and 1.5 AU withsmall eccentricities and inclinations (e ~ 0.01 and i ~ 0.05°).In the latter case, the initial embryos were those producedby the Weidenschilling et al. (1997) full terrestrial zone simu-lation of the midstage (see chapter by Kortenkamp et al.,2000). Giant planets were not included. Figure 3 shows theevolution of the mass distribution of embryos from one ofthe ACL99 simulations using the Weidenschilling et al. (1997)initial conditions. The general pattern of the dynamics issimilar in the simulations of both CW98 and ACL99, de-spite their use of different initial distributions of embryosand different numerical integration methods.

Figure 4 shows all the final planets from the 37 simula-tions performed in both CW98 and ACL99. Note that theACL99 simulations that began with the Weidenschilling etal. initial conditions considered a smaller total planetaryembryo mass (~1.8 M) than the CW98 runs, and thus yieldsomewhat smaller final planets. The most massive planetstend to form near or interior to 1 AU, where the surfacedensity and the collision frequency are the highest. All simu-lations of the late stage, whether direct integrations or MonteCarlo simulations, appear to produce planets with distribu-tions similar to Fig. 4 when considering a total mass ~2 Min the terrestrial region (see, e.g., Wetherill, 1992).


1.00M

ass

(Ear

th =

1)

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(f )

Semimajor Axis (AU) Semimajor Axis (AU)

Fig. 1. Masses and semimajor axes of the surviving objects at six different times from a simulation by Chambers and Wetherill (1998)that began with embryos in the terrestrial region and did not include Jupiter and Saturn (their model A). The horizontal line througheach symbol connects the perihelion and aphelion of the embryo’s orbit. Mutual interactions between embryos perturb them into crossingorbits and drive the accretion process until the final bodies’ orbits are well separated.


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Fig. 2. Same as Fig. 1, except this simulation included embryos in the present-day asteroid belt and Jupiter and Saturn (from Cham-bers and Wetherill, 1998, an example of their model C). In this case, the asteroid belt is cleared of embryos and the terrestrial regionhas a smaller number of final planets than in Fig. 1.


Initial condition

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Fig. 3. Masses and semimajor axes of the surviving objects at six different times from one of the simulations of Agnor et al. (1999,their Fig. 3). This simulation used as its starting condition the 22 largest embryos from the multizone embryo formation calculation ofWeidenschilling et al. (1997). Despite the different integration techniques and initial conditions, the evolution of the system is broadlysimilar to those shown in Figs. 1 and 2.


Figure 5 shows the time-averaged eccentricities and in-clinations of the final planets with masses greater than0.5 M from CW98 and ACL99. The average final eccen-tricity of the largest body from each simulation done byACL99 (who did not include Jupiter and Saturn) was 0.08,with time-averaged eccentricities all greater than 0.05.CW98 report comparable eccentricities for simulations per-formed with similar initial conditions. Even larger eccen-tricities resulted for runs that included Jupiter and Saturn;in this case the average eccentricity of the most massive finalplanet in each system was 0.18.

The large eccentricities of the planets produced by simu-lations also yields larger angular momentum deficits for thesystems formed. The angular momentum deficit, or AMD,is given by

AMD n a e ik k kk

N

k k= − −=

∑m 2

1

21 1 cos (2)

where N is the total number of planets, nk is the planet’smean-motion about the Sun, and mk is the planet’s mass.In the 10 simulations of ACL99, the average value of theAMD of the final planets formed was between 4.5 and 17times larger than that of the terrestrial planets, despite aninitial system angular momentum that exceeded the terres-

trial system by no more than 5%. Larger AMD values re-quire final planets more widely spaced (and therefore typi-cally fewer in number) than their terrestrial counterparts forsystem stability. In the ACL99 simulations that yielded twoadjacent planets with masses greater than 0.5 M (80% oftheir runs), the two massive planets had an average spacingof ⟨∆a⟩ = 44 RHill (with individual values ranging from 33to 55 RHill ), in comparison to the ∆a = 26.25 RHill separa-tion between Earth and Venus.

While the results obtained with various models of the latestage are thus in fairly close agreement, they continue toyield systems that are different in character than the cur-rent planets. An important question is why such discrepan-cies persist, even as modeling methods of embryo inter-actions have improved. The answer is likely that computa-tional and model limitations continue to restrict simulationsto simplified scenarios that neglect potentially influentialprocesses. For example, current simulations are not capableof handling the large numbers of smaller planetesimalspresent during the postrunaway evolution of planetary em-bryos, and perhaps during the final stage as well. Dynami-cal interactions with such a small body population mightyield lower planetary eccentricities, although this has yet tobe convincingly demonstrated. This issue is discussed inmore detail in section 4. However, assuming the basic con-

0

1

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3

ACL99CW98 Model ACW98 Model BCW98 Model C

Terrestrial Planets

0 1 2 3

Mas

s (m

/M )

Semimajor Axis (AU)

Fig. 4. Masses and semimajor axes of all of the final planets produced by the 27 simulations of Chambers and Wetherill (1998) andthe 10 simulations of Agnor et al. (1999). Initial conditions for models A and C are shown in Figs. 1a and 2a. Note that model B ofChambers and Wetherill (1998) had an identical distribution of initial embryos to that in model A, except that in model B Jupiter andSaturn were included in the simulation. The values for the terrestrial planets are included for comparison.


clusion that runaway growth yields embryos significantlysmaller than either Earth or Venus is correct, the final ac-cretion stage must have been characterized by large impactsbetween embryos. In section 3, we review the implicationsof such impacts for the final stages of growth of a terres-trial planet.

2.2. Radial Mixing During Late-Stage Accretion

The Earth, Moon, Mars, and meteorites originating in theasteroid belt display isotopic commonalities and differencesthat were generally thought to reflect their formation fromdistinct radial reservoirs of material in the protoplanetarydisk. For example, the O-isotopic signatures of terrestrialand lunar material fall on the same fractionation line, whichis distinct from that of meteorites believed to have origi-nated in the asteroid belt or Mars. The standard explana-tion is that the Earth and Moon formed from materialcontained in the same radial zone. This conceptually agreedwell with earlier views of planet formation, which employedthe concept of planetary growth from a local “feeding zone”(e.g., Safronov, 1969; Lewis, 1972). Recent studies suggestthat the concept of feeding zones may be relevant to the

midstages of planet formation, when eccentricities and in-clinations are low.

However, the evolution of a swarm of embryos into a fewplanets yields a large degree of mixing throughout the ter-restrial zone. Wetherill (1994) studied the initial location ofthe embryos that comprised the final planets that formed inhis Monte Carlo simulations. In general, all the embryosinitially in the region between 0.5 and 2.5 AU became ra-dially mixed, and the final planets were comprised of ma-terial that had its origins throughout this region. However,the average provenance of a planet was found to be some-what correlated to its semimajor axis, i.e., planets in theouter terrestrial region accreted more material from largerheliocentric distances than the inner planets. This generalresult is also evident in the recent N-body simulations, asshown in Fig. 6. CW98 find that final planets with a < 1 AUare comprised of material that originated primarily in the0.5–1.5-AU region (~75% of their final mass), but also con-tain material from further out in the protoplanetary disk(~25% from material originating between 2.5 and 3.5 AU). Itis thus consistent with the current accretion models that ter-restrial planets may “remember” the initial compositional zon-ing of the protoplanetary disk to a limited degree. Whether

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ACL99CW98 Model ACW98 Model BCW98 Model C

Earth and Venus

0.00 0.200.10 0.30 0.40

Incl

inat

ion

(deg

rees

)

Eccentricity

Fig. 5. The time-averaged inclinations and eccentricities of the final planets larger than 0.50 M are shown. Note the general ab-sence of planets with eccentricities of 0.03 or less. The current values for Earth and Venus are shown for comparison (squares). Time-averaged values are e ~ 0.03, and I ~ 2° for Earth and Venus.


such predictions can be reconciled with geochemical con-straints that appear to favor compositional zoning is animportant and open question.

3. LATE-STAGE IMPACT EVENTS

It is during the final stage that most of the end planetarycharacteristics — e.g., mass, spacing, orbital elements, spinangular momentum, presence of impact-generated satellites,etc. — are determined. For example, growing planets ac-quire their spins and obliquities as they accrete material,with the relative motions of the bodies at impact determin-ing the contributions of each accreted body to the spin an-gular momentum of the final planet. As a protoplanet grows,both the average impact velocity and average moment armof colliding bodies increase, so that the rotation state of aplanet is determined primarily during its final accretion (seechapter by Lissauer et al., 2000).

Large impact events could be either accretionary or ero-sive. The fate of debris generated during a planet-scale im-pact event would depend upon the specifics of the impact,e.g., impact angle, velocity relative to escape velocity, andmass ratio of impactor to target. The two-body escape ve-locity is

v G m m R Resc ≡ +( ) +( )2 1 2 1 2/ (3)

where R is radius. Collisions with vimp >> vesc would likelybe primarily erosive or even disruptive; such an impact hasbeen invoked to account for the loss of portions of Mer-cury’s early mantle, resulting in the planet’s current anom-

alously high density (e.g., Benz et al., 1988). Oblique im-pacts can result in the ejection of material into bound orbitaround the target protoplanet, which can then rapidly accreteto form satellites. Satellite formation via this mechanism hasbeen proposed for the origin of the Earth-Moon system(Hartmann and Davis, 1975; Cameron and Ward, 1976), thePluto-Charon binary (e.g., Stern et al., 1997, and referencestherein), and for asteroid-satellite systems (e.g., Durda,1996).

Determination of impact outcome as a function of spe-cific impact energy, material strength, and target size hasbeen the focus of a great body of experimental and theo-retical research. General scaling laws that predict quantitiessuch as crater size and total ejected mass exist for impactsthat span the size range from centimeter-sized particles upto target radii of about ~200 km (e.g., Housen and Hols-apple, 1990; Ryan and Melosh, 1998; Benz and Asphaug,1999). To date, numerical simulations of planet-scale im-pacts have focused primarily on the particular impact be-lieved responsible for the Earth-Moon system (Benz et al.,1986, 1987, 1989; Cameron and Benz, 1991). In general,an approximately Mars-sized body with an impact angularmomentum of at least the current angular momentum ofthe Earth-Moon system appears to be required to yield alunar-sized moon. However, general scaling relationships forplanet-scale impacts are just starting to be developed (e.g.,Canup et al., 2000, and discussion in section 4). Given thelack of such relationships, late-stage accretion simulationshave modeled collisional outcomes through the use ofsimple extrapolations from scaling laws derived for muchsmaller impacts (in the case of Wetherill’s Monte Carlosimulations), or by simply assuming that every impact re-sults in complete accretion (in the case of the N-body simu-lations).

3.1. Occurrence and Timing of Large Impacts

The Monte Carlo simulations of Wetherill (e.g., 1985,1986, 1992) were among the first to examine the predictedtiming and size of the largest impactors to collide with Earthduring its accretion (see also Hartmann and Vail, 1986).Wetherill (1985) performed three-dimensional simulationsthat each began with 500 bodies whose masses ranged fromabout 6 × 1024 to 1 × 1027 g, and with orbital distances be-tween 0.7 and 1.1 AU. In every case giant impacts (i.e.,impacts by objects with at least the mass of Mars) occurred,typically ~20 m.y. after the start of a simulation. In 1992,Wetherill performed similar calculations that began with aninitial distribution of embryos that spanned the entire ter-restrial region and the asteroid belt (0.4–3.8 AU). Similarnumbers of giant impacts occurred, but at somewhat latertimes of 5 × 107–108 yr. In both studies, about one impactonto a large planet by a body at least as massive as Marsoccurred for each simulation that contained a final planetwith mass similar to that of Venus or Earth.

ACL99 characterized the collisions in their N-body simu-lations in terms of impactor mass, velocity, and impact an-

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tion

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Fig. 6. The fraction of the planet mass obtained from differentregions of the protoplanetary disk are shown for the final planetsof the simulations presented in Chambers and Wetherill (1998,their Fig. 21). The three curves are for planets with final semima-jor axes of 0 < a < 1 AU, 1 < a < 2 AU, and 2 < a < 3 AU. Whileplanets form from some material near their end orbital radius,radial mixing of material causes planets to acquire a significantfraction of their mass from more distant regions.


gular momentum as a function of time (see Fig. 7). The im-pactor was defined to be the less massive member of a col-liding pair. Figure 8 gives a breakdown of the impactvelocity and collision angular momentum as a function ofimpactor mass from the same simulations. The initial em-

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Time (yr)

Fig. 7. The impactor mass, impact velocity, and the angular momentum of the collisional encounter are shown as a function of thetime at which each impact occurred. All collisions from the 10 simulations of Agnor et al. (1999, their Fig. 8) are shown. The angularmomentum of the Earth-Moon system (L–M) is shown by the dashed line. Simulations 1–8 utilized the Weidenschilling et al. (1997)embryos as their starting conditions.

bryos considered in ACL99 had masses of 0.04–0.15 M.As a simulation proceeds, embryos collide and merge, anda spectrum of embryo masses develops. A weak degree ofdynamical friction (see discussion in chapter by Kortenkampet al., 2000) causes the smaller bodies in the disk to obtain


Fig. 8. The impact velocity and angular momentum of the collisional encounter are shown as a function of the impactor mass for thesame collisions displayed in Fig. 7 (Agnor et al., 1999, their Fig. 9). Note that some small, high-velocity impactors are capable ofdelivering more than 1 L–M to the system.

larger eccentricities and inclinations (and thus higher im-pact velocities) than their more massive counterparts. Aftera few million years (at which time the embryo masses spanabout a factor of five), some impacts occur at several timesthe escape velocity (see Fig. 7b).

These results generally resemble those of Wetherill (1992)in terms of the timing of the largest impacts. The ACL99final planets with m > 0.5 M experienced their largestimpact at an average time of 29 m.y. (with individual val-ues ranging from 1.4 to 95 m.y.); the last impact onto theseplanets occured at an average time of 46 m.y. (with indi-vidual values ranging from 3.1 to 108 m.y.). These timingsare in relatively good agreement with isotopic constraintson the age of the Earth and Moon. The Hf-W chronometerplaces the time of core formation in the Earth and Moon at~50 m.y. (e.g., Lee et al., 1997; see chapter by Halliday etal., 2000), while the formation interval for Earth based on theterrestrial excess of the heaviest isotopes of Xe is ~100 m.y.(see chapter by Podosek and Ozima, 2000).

In an attempt to identify impacts that could result in theformation of a lunar-sized satellite, ACL99 described thosecollisions with an encounter angular momentum equal toor exceeding the current angular momentum of the Earth-Moon system (L–M) as “potential moon-forming impacts.”This classification reflects the very rudimentary knowledgeof the impact dynamics required to form an impact-gener-ated satellite, and is somewhat different than the “giantimpacts” of Wetherill (1985, 1992), which were classifiedby the impactor mass. In the 10 ACL99 accretion simula-tions, there were a total of 20 potential moon-forming im-pacts (see Figs. 7b and 8a). In addition, 25% of the finalplanets larger than 0.50 M experienced more than one suchimpact.

In the ACL99 simulations that followed the evolution ofthe 22 embryos from Weidenschilling et al. (1997), thenumber of moon-forming collisions that occurred in eachsimulation ranged from 0 to 3, with an average of about 1.6per simulation. In the simulations that began with 50 initial

0

2

4

6

8

0.00 0.10 0.20 0.40

(a)

(b)

Vim

p/V

esc

0

1

2

3

4

0.00 0.10 0.20

0.30

0.30 0.40

Lim

p/L

–

M

Simulations 1–8Simulations 9–10

Impactor Mass (m/M )

Impactor Mass (m/M )


embryos (and a slightly larger total embryo mass of 2 M),there were an average of 3.5 moon-forming collisions persimulation. The impact velocities of the moon-forming col-lisions were typically somewhat larger than the the two-bodyescape velocity. With the exception of one (0.04 M) im-pactor that had an impact velocity of 3.44 vesc, the Limp >L –M collisions occurred with velocities in the range 1.00–1.63 vesc, with an average of 1.20 vesc (see Fig. 8a). High-resolution simulations of the moon-forming impact per-formed to date (e.g., Cameron, 1997; Cameron, 2000) as-sume vimp = vesc, and so have considered only the lower limitof possible specific impact energies.

3.2. Following the Growth of a Planet

The spin angular momentum of an embryo will evolvedue to each of the large impacts it experiences. As a basicmodel for this process, ACL99 assumed that for each colli-sion, the spin angular momentum of the merged body wasjust the sum of the spin angular momenta of the two col-liding bodies and the orbital angular momentum of the twobodies about their center of mass. The rotational periods ofthe growing planets were then calculated by assuming thatall bodies were spheres of uniform density. In general, theaccretion of material and angular momentum was likely lessthan 100% efficient during collisions between planetaryembryos, and the assumption of inelastic mergers to modelcollisions will tend to overestimate both of these quantities.In addition, no account was made for the precession of spinaxes that will result for oblate planets, or for the evolutionof planetary obliquities due to spin-orbit coupling (see chap-ter by Williams and Pollard, 2000).

Figure 9 shows the evolution of the mass, the magnitudeof the spin angular momentum (in units of L–M), obliq-uity, and rotation period of a typical Earth-like planet (de-fined to be a planet with m ≥ 0.50 M ). In this particularcase, each collision results in a net increase in the magni-tude of the spin angular momentum of the planet. Impactsreorient the direction of the spin angular momentum vec-tor, and in general, the spins and obliquities of the growingplanets during the late stage are quite large (see chapter byLissauer et al., 2000). For collisions with vimp = vesc, theangular momentum of the impact, Limp, scaled by the criti-cal angular momentum for rotational stability, Lcrit, is just

L

L Kc c c cimp

criti i i i= −( ) ( )+ −2

1 11/3 1/3sinξ (4)

with

L M K Gcrit T≡ 5/3 1/63 4/( )πρ (5)

where MT and RT is the total colliding mass and the radiusof the combined body, ρ and K are the density and gyra-tion constants of the colliding bodies, ci is the fraction ofthe total mass contained in the smaller of the colliding bod-ies, K = I/MTRT

2 where I is the moment of inertia, and ξ isthe impact angle. The angular momentum imparted from asingle collision can approach or exceed Lcrit for large ci. For

example, with like-sized bodies (and assuming a gyrationconstant equal to that of Earth, or K = 0.335), Limp/Lcrit =1.33 sin ξ. Indeed, the ACL99 simulations find planetaryrotation rates that often exceed the rotational stability limit.Clearly the assumption of merger in such cases is invalid,as the excess angular momentum would instead be carriedaway in ejected debris or in debris placed into circumplane-tary orbit.

Even small embryo collisions (mimp ≈ 0.04 M ) can makeangular momentum contributions to a planet that are sig-nificant in comparison to L–M. In Fig. 10 the mass and themagnitude of the spin angular momentum of a 0.80 Mplanet from one of the ACL99 simulations are shown as afunction of time. This planet experiences two collisions withinitial embryos (at t ~ 32 × 106 and ~50 × 106 yr), whichboth act to reduce the magnitude of the planet’s spin angu-lar momentum, slowing its spin rate. The second of thesecollisions occurred with vimp = 1.4 vesc, and had a collisionalangular momentum of 0.55 L–M. The largest impact tostrike this planet was also the last impact the planet experi-enced, which was the case for about one-half the Earth-like

0.0

0.2

0.4

0.6

0.8

1.0

0 2.0 × 1071.0 × 107 3.0 × 107 5.0 × 1074.0 × 107

Impa

ctor

Mas

s (m

/M )

Obl

iqui

ty (

degr

ees)

Rot

atio

n P

erio

d (h

r)

Time (yr)

0

1

2

3

0 2.0 × 1071.0 × 107 3.0 × 107 5.0 × 1074.0 × 107

Time (yr)

150

100

50

0

0 2.0 × 1071.0 × 107 3.0 × 107 5.0 × 1074.0 × 107

Time (yr)

0

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0 2.0 × 1071.0 × 107 3.0 × 107 5.0 × 1074.0 × 107

Time (yr)

Rotationally unstable

Lsp

in/L

–

M

Fig. 9. The mass, magnitude of the spin angular momentum inunits of the Earth-Moon system, obliquity, and rotation period ofa typical planet (the 0.73 M final planet shown in Fig. 3) are plot-ted as a function of time (Agnor et al., 1999, their Fig. 10). Thedotted line of the rotation period vs. time graph indicates the ro-tational stability limit. The small oscillations in the obliquity aredue to the motion of the body’s orbit; the spin axis of the body wasassumed to remain fixed in the inertial frame between collisions.


0.0

0.2

0.4

0.6

0.8

1.0

0 4 × 1072 × 107 6 × 107 1 × 1088 × 107

Mas

s (m

/M )

Time (yr)

0.0

0.5

1.0

1.5

2.0

2.5

0 4 × 1072 × 107 6 × 107 1 × 1088 × 107

Time (yr)

Lsp

in/L

–

M

Fig. 10. The mass and magnitude of the spin angular momen-tum in units of the Earth-Moon system are shown as a functionof time for a planet from a simulation that began with smaller butmore closely spaced initial embryos (Agnor et al., 1999, theirFig. 13). Small impacts (~0.04 M), such as those occurring at3.2 × 106 yr and 5.0 × 107 yr, can make significant contributionsto the angular momentum accretion of the planets.

planets in the ACL99 simulations. During the growth of theplanet shown in Fig. 11, the largest impact (at t = 13 ×106 yr) was followed by two smaller impacts, the last ofwhich resulted in a net decrease of 0.85 L–M in the mag-nitude of the spin angular momentum. Approximately 40%of the Earth-like planets produced by the ACL99 simula-tions experienced collisions that decreased the magnitudeof the spin angular momentum of the planet by 0.10 L–Mor more.

For the Earth-like planets produced in ACL99, the larg-est and second largest impactor contributed an average of 30%

0.0

0.2

0.4

0.6

0.8

1.0

0 2.0 × 1071.0 × 107 3.0 × 107 5.0 × 1074.0 × 107

Mas

s (m

/M )

Time (yr)

0

1

2

3

4

0 2.0 × 1071.0 × 107 3.0 × 107 5.0 × 1074.0 × 107

Time (yr)

Lsp

in/L

–

M

Fig. 11. The planet mass and spin angular momentum as a func-tion of time for a planet that experienced a net reduction in itsspin angular momentum due to cancellation between contributionsfrom multiple impacts (Agnor et al., 1999, their Fig. 11).

and 19% of the final planet mass and contributed an aver-age of 1.44 L–M and 0.67 L–M respectively. For nearly50% of these planets, the last impactor was also the larg-est. However, for an equal number of final planets, the lastimpactor was less massive than both the largest and second-largest impactor. In these cases, the last impact contributedan average of only 8.3% of the final planet mass, but con-tributed an average angular momentum of 0.76 L–M withvalues ranging from 0.10 to 1.83 L–M. Furthermore, theseplanets accreted an average of 31% (range 7–62%) of theirfinal planet mass after experiencing their first Limp ≥ L –Mimpact. These statistics were derived from a limited num-ber of simulations; however, they suggest that the spin an-gular momentum of a terrestrial planet is likely the resultof more than one large impact, and that moon-forming col-lisions can occur before planetary growth is completed.

3.3. Implications for Lunar Origin

Two fundamental constraints on models of the formationof the Earth-Moon system are the masses of the bodies andthe current system angular momentum. Models of lunarformation have generally made the assumption that theMoon-forming impact was the last impact Earth experi-enced. However, recently this constraint has been relaxedin an order to yield a lunar-mass Moon as described below.

SPH (“smoothed-particle hydrodynamics”) simulationsof the lunar-forming impact have typically considered col-lisions with impactor to target mass ratios of 3:7 and 2:8,no initial spin angular momentum of the impactor or target,and vimp = vesc (Cameron, 1997; see also chapter by Cam-eron, 2000). These studies have investigated a variety ofimpact angular momenta and system masses (0.5–1.0 M).Recent N-body simulations of the accumulation of the Moonfrom an impact-generated disk indicate that a disk mass ofat least two lunar masses is required to form the Moon (e.g.,Ida et al., 1997; see chapter by Kokubo et al., 2000). Theonly impacts that SPH simulations have shown to be capableof producing a circumterrestrial disk massive enough toform the Moon are those with a total mass of 1.0 M andan impact angular momentum greater than 2.0 L–M, orthose with a total mass of ~0.6 M and 1.0 L –M (Cam-eron and Canup, 1998; Canup et al., 2000). Each of thesescenarios is unable to simultaneously account for both themass and angular momentum of the Earth-Moon systemwith a single impact. The first case, with Limp ~ 2 L –M,requires that the Earth-Moon system somehow rid itself ofan angular momentum excess ~1 L–M after the moon-form-ing event. As solar tides could remove only a small frac-tion of this, this scenario would seem to require that Earthexperienced another impact to reset Earth’s spin. In thesecond case, Earth acquires another ~0.3–0.4 M of mate-rial after the Moon has formed. This scenario also requiresthat Earth experienced later impacts.

The results of the recent late-stage simulations show thatseveral large impacts typically affect a planet’s end state,implying that Moon-forming impacts with total masses andangular momenta different from that of the Earth-Moon


system may be plausible. The ACL99 simulations find thatplanets occasionally experienced glancing collisions at sev-eral times the escape velocity with smaller (~0.05 M) im-pactors that were capable of delivering up to 1.1 L–M. Atpresent it isn’t known whether or not this type of collisionwould be efficient at producing a circumplanetary disk. Ifthis type of collision occurred after the Moon had alreadyformed, it could conceivably reset the spin angular momen-tum of Earth, possibly offering a resolution to the highangular momentum impact scenario described above. Thelate-stage simulations also predict that embryos have rapidspin rates throughout the final accumulation stage. In par-ticular, the final planets of the ACL99 simulations that ex-perienced Limp ≥ L –M impacts had an average angularmomentum of ~0.91 L–M (an average rotational period of2.4 hr) immediately prior to these collisions. While the fu-ture refinements of planet formation models (e.g., inclusionof the effects of small bodies during the final accretionstage) may modify this estimate, it is significant enough tosuggest that the spin of Earth prior to the Moon-formingevent was likely not negligible, and that Moon-formingcollisions in which the impactor and/or the target are spin-ning should be studied. Perhaps, e.g., an Earth with a pro-grade spin prior to the lunar-forming impact would resultin higher yields of bound orbiting debris for a given im-pact angular momentum. Finally, the late-stage simulationsindicate that large impacts occur with vimp > vesc, and so suchcollisions should also be explored in the context of the lu-nar-forming event. Higher specific impact energies (∝ v2

imp)might act to increase the yield of ejected material for a givenimpact angular momentum (∝ vimp), and potentially help tomitigate the difficulty to date in forming a massive proto-lunar disk with an Limp ~ L –M impact.

3.4. Implications for the Frequency ofImpact-triggered Satellites

To some extent, the early perception that large impactevents were rare coincided well with the fact that among thecurrent terrestrial planets, only Earth has a sizeable, impact-generated moon. However, the accretion simulations per-formed over the past decade depict a terrestrial planet forma-tion process whose final stages are dominated by stochasticimpact events. In fact, it seems likely that terrestrial plan-ets may have experienced multiple giant impacts in theirhistories, and that many such impacts may have led to theformation of satellites (e.g., Canup et al., 2000).

If accretion models now suggest that most terrestrialplanets undergo such impact events, is it not a contradic-tion that, e.g., Venus does not currently possess a satellite?Not necessarily. First, while giant impacts appear to havebeen commonplace in the inner solar system, there are stillonly certain impact orientations that have been shown toyield significant amounts of debris in circumplanetary or-bit. So while oblique, relatively low-velocity collisionsmight generate a circumplanetary disk, head-on collisions,

near misses, or collisions with vimp >> vesc might generatedebris that reimpacts the planet or escapes entirely to he-liocentric orbit. Large impacts may be common, but satel-lite-generating impacts will be less so.

Second, it is also plausible that other terrestrial planetsonce had impact-generated satellites that simply did notpersist against continuing bombardment during the late stageor against orbital evolution due to tidal interaction. A simplecriterion for satellite stability is that the orbit initially ex-pands rather than contracts as it tidally evolves, or a > aco,where a is the initial semimajor axis of the satellite, and acois the co-rotation radius

aGM

cop≡

ω2

1/3

(6)

where Mp and ω are the mass and angular velocity of theplanet. Since an impact-generated satellite cannot conceiv-ably accrete within the Roche radius (see chapter by Kokuboet al., 2000), a ≥ aRoche, with

a RRoches

p≈ 2 51/3

.ρ

pρ(7)

where ρs is the satellite density, and ρp and Rp are the den-sity and radius of the planet. The requirement that aRoche >aco then yields a constraint on the initial spin rate of theplanet

ω π ρo sG≥ 4/3

2 5 3

1/2

( . )(8)

For a satellite density equal to that of the Moon, a planet’sday must be shorter than about 7 hr for co-rotation to fallwithin the Roche limit. Slower rotators would quickly losetheir impact-generated moons to inward tidal decay on veryshort timescales (e.g., approximately years for an Earth-sized planet and a lunar-mass satellite with a = 3 R). Thelong-term survival of a satellite that initially forms outsidethe co-rotation radius is not guaranteed, and will depend onthe interplay of solar and satellite tides, which both act toslow the planet’s rotation and cause aco to evolve outward(Burns, 1973; Ward and Reid, 1973). If aco overtakes a, theeventual fate of the satellite will again be a collision withthe planet (although in this case the timescale for the or-bital decay can be much longer).

4. OPEN ISSUES

Models of the final stage of terrestrial accretion are gen-erally successful, producing a few planets from a system ofplanetary embryos in a timeframe consistent with isotopicconstraints for the age of the Earth and Moon. Some dis-parities between current predictions and the terrestrial plan-ets persist, in particular with regard to accounting for the


very low eccentricities of Earth and Venus. However, suchdifferences are likely the result of simplified assumptionsemployed by the late-stage models, which to date have allignored the production and dynamical influence of smallmaterial. From studies of the runaway growth phase, it iswell known that the general role of a background popula-tion of small bodies is to decrease the relative velocities (andtherefore eccentricities and inclinations) of the largest bod-ies via dynamical friction. A key open question is then howthe presence of a planetesimal swarm will alter the dynami-cal environment in which embryos accrete to form planets.Whether the influence of small bodies in the late stage couldbe sufficient to yield Earth-like planets with nearly circularorbits remains uncertain, and addressing this question willrequire an improved understanding of both the generationof debris during planet-scale collisions, and the persistenceof a background population as it co-evolves with a systemof embryos.

The outcomes of planet-scale collisions are still poorlyunderstood, and the type of hydrodynamic simulations uti-lized to model the lunar-forming impact could be extendedto a broader range of collisional parameter space in orderto better generalize impact outcomes. Accounting for theamount of ejected debris, its velocity, and its angular mo-mentum will affect estimates of both the potential back-ground population of small material and planetary spin rates.Current scaling relationships derived for meter- to 100-ki-lometer-sized targets relate QD

*, the critical specific energyrequired to disperse one-half the target mass, to target size(e.g., Benz and Asphaug, 1999). Given these relationshipsfor QD

*, and assuming a linear relationship between the spe-cific impact energy, Q, and the amount of material ejected(as implied in Love and Ahrens, 1996), ACL99 estimatedthe fraction of material that would be dispersed for each ofthe collisions in their simulations. This fraction ranged from~1% to 20% of the total mass involved in a collision (theirFig. 15). SPH simulations of the lunar-forming impact (Cam-eron, 2000; Canup et al., 2000) suggest that the amount ofejected mass will also depend upon target-to-impactor massratio and impact angular momentum, in addition to the spe-cific impact energy.

Given that some nonneglible fraction of material isejected during collisions between planetary embryos, thequestion is then how long this material will persist as abackground population during the final stage. Much of thedebris ejected in an impact might rapidly reaccumulate ontoits parent body during subsequent orbital passes, unless itis dynamically perturbed by other embryos or the giant plan-ets. If an impact was in fact responsible for the loss of muchof Mercury’s mantle, we have at least one example case inwhich the great majority of the ejected material must nothave reaccreted onto the source planet. Assuming immedi-ate reaccretion is avoided, the timescales for sweep-up ofthe remaining debris could be comparable to (or longer than)the time between embryo collisions. A simple fragmenta-tion model, which assumed that a few percent of the mass

involved in each collision was ejected as fragments, wasincluded in two-dimensional N-body simulations of theaccretion of embryos in the terrestrial region by Alexanderand Agnor (1998). Interestingly, that work found that theejected fragments quickly contained up to 50% of the totalsystem mass. Another possible contribution to a late-stagebackground population could be leftover remants from theearlier runaway stage. The Weidenschilling et al. (1997)simulation did not include fragmentation, and its predictionthat 10% of the system mass remained in small bodies af-ter a million years should therefore be viewed as a lowerlimit. Recent orbit integrations of test particles placedthroughout the current terrestrial region (see Fig. 5 in chap-ter by Hartmann et al., 2000) find that near 50% of theparticles remain in the 0.7–1.3-AU region after 100 m.y.

Modeling the possible interactions between a system ofembryos and a background population of small material fortimescales ~108 yr presents many computational and algo-rithmic challenges for the next generation of planet forma-tion simulations. An important step in this regard will in-volve an improved understanding of the transition from therunaway growth period to the late stage. While the embryosin the Weidenschilling et al. (1997) simulations (which con-tained about 90% of the system mass) remained in low-eccentricity, stable orbits for times longer than 105 yr, theevolution of the same embryos using direct integrations inACL99 (but ignoring the 10% of mass contained in smallermaterial) destabilized on times ~104 yr. Determining thesource of these differences will be closely related to effortsto determine the role of small material in late-stage accre-tion.

While important processes relevant to the final accumula-tion of the terrestrial planets are thus not yet completelyunderstood, the basic predictions of existing models are infairly good agreement with the current terrestrial planets.Additional attention to issues neglected in simulations per-formed to date may help to resolve the outstanding ques-tion of forming low-eccentricity planets. But in general, astochastic phase dominated by large impacts appears aninevitable consequence of the evolution of tens to hundredsof planetary embryos into a final few planets. Barring amajor change in our understanding of the outcome of thepreceeding runaway growth stage, this conclusion appearsrobust, and will likely persist even as late-stage accretionmodels are further refined.

Acknowledgments. The authors wish to acknowledge help-ful reviews by G. Wetherill and J. Chambers, and support fromNASA’s Origins of Solar Systems and Graduate Student Research-ers programs.

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accretion of the terrestrial planets and the earth-moon system

Documents

terrestrial planets

stage of growth

marssized planetary

formation of kilometer

manyplanetary embryos

asystem of embryos

gplanetary embryos

final planets